Solutions Manual Introduction to Finite Elements in Engineering 4th Edition Tirupathi R. Chandrupatla, Ashok D. Belegund...
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CONTENTS Preface Chapter 1
Matrix Algebra and Gaussian Elimination
Beams and Frames
Two-Dimensional Problems Using Constant Strain Triangles
Axisymmetric Solids Subjected to Axisymmetric Loading
Two-Dimensional Isoparametric Elements and Numerical Integration
Three-Dimensional Problems in Stress Analysis
Scalar Field Problems
Preprocessing and Postprocessing
PREFACE This solutions manual serves as an aid to professors in teaching from the book Introduction to Finite Elements in Engineering, 4 th Edition. The problems in the book fall into the following categories: 1. 2. 3. 4.
Simple problems to understand the concepts Derivations and direct solutions Solutions requiring computer runs Solutions requiring program modifications
Our basic philosophy in the development of this manual is to provide a complete guidance to the teacher in formulating, modeling, and solving the problems. Complete solutions are given for problems in all categories stated. For some larger problems such as those in three dimensional stress analysis, complete formulation and modeling aspects are discussed. The students should be able to proceed from the guidelines provided. For problems involving distributed and other types of loading, the nodal loads are to be calculated for the input data. The programs do not generate the loads. This calculation and the boundary condition decisions enable the student to develop a physical sense for the problems. The students may be encouraged to modify the programs to calculate the loads automatically. The students should be introduced to the programs in Chapter 12 right from the point of solving problems in Chapter 6. This will enable the students to solve larger problems with ease. The input data file for each program has been provided. Data for a problem should follow this format. The best strategy is to copy the example file and edit it for the problem under consideration. The data from program MESHGEN will need some editing to complete the information on boundary conditions, loads, and material properties. We thank you for your enthusiastic response to our first three editions of the book. We look forward to receive your feedback of your experiences, comments, and suggestions for making improvements to the book and this manual. Tirupathi R. Chandrupatla P.E., CMfgE Department of Mechanical Engineering Rowan University, Glassboro, NJ 08028 e-mail: [email protected]
Ashok D. Belegundu Department of Mechanical and Nuclear Engineering The Pennsylvania State University University Park, PA 16802 e-mail: [email protected]
CHAPTER 1 FUNDAMENTAL CONCEPTS 1.1
We use the first three steps of Eq. 1.11
Adding the above, we get
2(1 + v) Above relations can be written in the form P = DS where D is the material property matrix defined in Eq. 1.15.
Note that u 2 (x) satisfies the zero slope boundary condition at the support.
Contours of sx Contours of sy and y xy are obtained by changing Z in the script file. The numbers on the contours show the function values.
The maximum value of sx is at any of the corners of the square region. The maximum value is 21 . ■